Advanced Macroeconomics 6. Rational Expectations and Consumption Karl Whelan School of Economics, UCD Spring 2020 Karl Whelan (UCD) Consumption Spring 2020 1 / 28
A Model of Optimising Consumers We will now move on to another example involving the techniques developed in the last topic. Here, we will look at the question of how a consumer with rational expectations will plan their spending over a lifetime. Along the way, we will Discuss budget constraints and wealth accumulation. 1 Show how consumption depends on net wealth and expectations of 2 future income. Illustrate some pitfalls in using econometrics to assess the effects of 3 policy. Discuss the link between consumption spending and fiscal policy. 4 Discuss the link between consumption spending and the return on 5 various financial assets. Karl Whelan (UCD) Consumption Spring 2020 2 / 28
The Household Budget Constraint Let A t be household assets, Y t be labour income, and C t stand for consumption spending. Stock of assets changes by A t +1 = (1 + r t +1 ) ( A t + Y t − C t ) where r t +1 is the return on household assets at time t + 1. Note that Y t is labour income (income earned from working) not total income because total income also includes the capital income earned on assets (i.e. total income is Y t + r t +1 A t .) This can be written as a first-order difference equation in our standard form A t +1 A t = C t − Y t + 1 + r t +1 Assume that agents have rational expectations and that return on assets equals a constant, r : 1 A t = C t − Y t + 1 + r E t A t +1 Karl Whelan (UCD) Consumption Spring 2020 3 / 28
The Intertemporal Budget Constraint We have another first-order stochastic difference equation 1 A t = C t − Y t + 1 + r E t A t +1 Using the same repeated substitution method as before, we get ∞ E t ( C t + k − Y t + k ) � A t = (1 + r ) k k =0 We are assuming E t A t + k (1+ r ) k goes to zero as k gets large. One way to understand this equation is to re-writing it as ∞ ∞ E t C t + k E t Y t + k � � (1 + r ) k = A t + (1 + r ) k k =0 k =0 This is called the intertemporal budget constraint . The present value sum of current and future household consumption must equal the current stock of financial assets plus the present value sum of current and future labour income. Karl Whelan (UCD) Consumption Spring 2020 4 / 28
Piketty and r > g Karl Whelan (UCD) Consumption Spring 2020 5 / 28
Piketty’s Conjecture Piketty’s most famous conjecture is there is a natural tendency in capitalist economies for wealth to accumulate faster than income. This conjecture can be understood on the basis of the simple budget identity we are working with here. A t +1 = (1 + r ) ( A t + Y t − C t ) What is the growth rate of the stock of assets? = r + (1 + r ) ( Y t − C t ) A t +1 − A t A t A t This means the growth rate of assets equals r plus an additional term that will be positive as long as Y t > C t i.e. as long as labour income is greater than consumption. Piketty points out that the rate of return on assets r has tended historically to be higher than the growth rate of GDP, which he terms g . If C t < Y t assets would grow at a rate greater than r which is greater than g . However, it is also possible to have C t > Y t and still have assets growing at a rate smaller than r but greater than g . Karl Whelan (UCD) Consumption Spring 2020 6 / 28
When Do Assets Grow Faster Than Income? Answer: Assets grow faster then income when g < r + (1 + r ) ( Y t − C t ) A t This can be re-arranged to give C t − Y t < r − g A t (1 + r ) So assets will grow faster than incomes if the amount of capital income consumed as a share of total assets is below a specific value. Is there any result in economics that leads us to believe that this inequality should generally hold? Not to my knowledge but perhaps it is more likely when r rises well above g . Piketty perhaps overstates the extent to which, on its own, the fact that r > g is a “fundamental force for divergence.” Rising assets relative to income and growing inequality of wealth is likely driven by other forces making income distribution more unequal and reducing share of income going to workers rather than being related to some innate “law of capitalism” that drives wealth up at faster pace than incomes. Karl Whelan (UCD) Consumption Spring 2020 7 / 28
Optimising Consumers We will assume that consumers wish to maximize a welfare function of the form ∞ � k � 1 � W = U ( C t + k ) 1 + β k =0 where U ( C t ) is the instantaneous utility obtained at time t , and β is a positive number that describes the fact that households prefer a unit of consumption today to a unit tomorrow. If the future path of labour income is known, consumers choose a path for consumption to maximise the following Lagrangian: � k � � ∞ ∞ ∞ � 1 Y t + k C t + k � � � L = U ( C t + k ) + λ A t + (1 + r ) k − (1 + r ) k 1 + β k =0 k =0 k =0 For every current and future value of consumption, C t + k , this yields a first-order condition of the form � k � 1 λ U ′ ( C t + k ) − (1 + r ) k = 0 1 + β Karl Whelan (UCD) Consumption Spring 2020 8 / 28
Consumption Euler Equation For k = 0, this implies U ′ ( C t ) = λ For k = 1, it implies � 1 + β � U ′ ( C t +1 ) = λ 1 + r Putting these two equations together, we get � 1 + r � U ′ ( C t ) = U ′ ( C t +1 ) 1 + β When there is uncertainty about future labour income, this optimality condition can just be re-written as � 1 + r � U ′ ( C t ) = E t [ U ′ ( C t +1 )] 1 + β This implication of the first-order conditions for consumption is sometimes known as an Euler equation . Karl Whelan (UCD) Consumption Spring 2020 9 / 28
The Random Walk Theory of Consumption In an important 1978 paper, Robert Hall discussed a specific case of the consumption Euler equation. He assumed aC t + bC 2 U ( C t ) = t = r β In this case, the Euler equation becomes a + 2 bC t = E t [ a + 2 bC t +1 ] Thus which simplifies to C t = E t C t +1 Because, the Euler equation holds for all time periods, we have C t = E t ( C t + k ) k = 1 , 2 , 3 , ... All future expected values of consumption equal the current value. Because it implies that changes in consumption are unpredictable, this is sometimes called the random walk theory of consumption. Karl Whelan (UCD) Consumption Spring 2020 10 / 28
The Rational Expectations Permanent Income Hypothesis Consumption changes are unpredictable but what determines the level of consumption each period? Insert E t C t + k = C t into the intertemporal budget constraint to get ∞ ∞ C t E t Y t + k � � (1 + r ) k = A t + (1 + r ) k k =0 k =0 Now we can use the geometric sum formula to turn this into a more intuitive formulation: ∞ 1 1 = 1 + r � (1 + r ) k = 1 1 − r 1+ r k =0 So, Hall’s assumptions imply the following equation, which we will term the Rational Expectations Permanent Income Hypothesis : ∞ r r E t Y t + k � C t = 1 + r A t + (1 + r ) k 1 + r k =0 Karl Whelan (UCD) Consumption Spring 2020 11 / 28
Implications of RE-PIH The Rational Expectations Permanent Income Hypothesis ∞ r r E t Y t + k � C t = 1 + r A t + 1 + r (1 + r ) k k =0 states that the current value of consumption is driven by three factors: The expected present discounted sum of current and future labour 1 income. The current value of household assets. This “wealth effect” is likely to 2 be an important channel through which financial markets affect the macroeconomy. r The expected return on assets: This determines the coefficient, 1+ r , that 3 multiplies both assets and the expected present value of labour income. In this model, an increase in this expected return raises this coefficient, and thus boosts the propensity to consume from wealth. Karl Whelan (UCD) Consumption Spring 2020 12 / 28
An Example: Constant Expected Growth in Income Suppose households expect labour income to grow at a constant rate g : E t Y t + k = (1 + g ) k Y t This implies � k ∞ 1 + r A t + rY t r � 1 + g � C t = 1 + r 1 + r k =0 As long as g < r (and we will assume it is) then we can use the geometric sum formula to simplify this expression � k ∞ � 1 + g 1 = 1 + r � = 1 − 1+ g 1 + r r − g 1+ r k =0 This implies a consumption function of the form r r C t = 1 + r A t + r − g Y t Note that the higher is expected future growth in labour income g , the larger is the coefficient on today’s labour income and thus the higher is consumption. Karl Whelan (UCD) Consumption Spring 2020 13 / 28
A Warning About Econometrics and Policy Evaluation Consider an economy where households have always expected their after-tax labour income to grow at rate g . Now suppose the government decide to introduce a one-period income tax cut that boosts after-tax labour income by one unit. They ask an econometrician to figure out how much this will raise consumption. The econometrican goes to the data which previously has been characterised by r r C t = 1 + r A t + r − g Y t r and says the answer is r − g . In reality, that relationship only works when people expect labour income growth of g and that won’t hold anymore when there is a once-off temporary tax cut. The true model is still ∞ r r E t Y t + k � C t = 1 + r A t + (1 + r ) k 1 + r k =0 r so consumption will only go up by 1+ r . Karl Whelan (UCD) Consumption Spring 2020 14 / 28
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